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Rate of the enhanced dissipation for the two-jet Kolmogorov type flow on the unit sphere

We study the enhanced dissipation for the two-jet Kolmogorov type flow which is a stationary solution to the Navier-Stokes equations on the two-dimensional unit sphere given by the zonal spherical harmonic function of degree two. Based on the pseudospectral bound method developed by Ibrahim, Maekawa, and Masmoudi [15] and a modified version of the Gearhart-Prüss type theorem shown by Wei [48], we derive an estimate for the resolvent of the linearized operator along the imaginary axis and show that a solution to the linearized equation rapidly decays at the rate $O(e^{-\sqrtν\,t})$ when the viscosity coefficient $ν$ is sufficiently small as in the case of the plane Kolmogorov flow.